3.111 \(\int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=541 \[ \frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (2 b e-a f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{e} \sqrt{a+b x^2}}\right ),\frac{e (b c-a d)}{c (b e-a f)}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b (c f+d e)) \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \]

[Out]

(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*Sqrt[a + b*x^2]) - (Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a
+ b*x^2))]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((
b*c - a*d)*e)])/(2*b*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]) + ((b*c - a*d)*Sqrt[e]*(2*b*e - a*
f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a
+ b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(2*b^2*c*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*S
qrt[e + f*x^2]) - (a*(a*d*f - b*(d*e + c*f))*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[
(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(2
*b^2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.436035, antiderivative size = 541, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206, Rules used = {556, 554, 424, 552, 419, 553, 537} \[ \frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (2 b e-a f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b (c f+d e)) \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/Sqrt[a + b*x^2],x]

[Out]

(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*Sqrt[a + b*x^2]) - (Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a
+ b*x^2))]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((
b*c - a*d)*e)])/(2*b*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]) + ((b*c - a*d)*Sqrt[e]*(2*b*e - a*
f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a
+ b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])/(2*b^2*c*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*S
qrt[e + f*x^2]) - (a*(a*d*f - b*(d*e + c*f))*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[
(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(2
*b^2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])

Rule 556

Int[(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2])/Sqrt[(e_) + (f_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt
[a + b*x^2]*Sqrt[c + d*x^2])/(2*Sqrt[e + f*x^2]), x] + (Dist[(e*(b*e - a*f))/(2*f), Int[Sqrt[c + d*x^2]/(Sqrt[
a + b*x^2]*(e + f*x^2)^(3/2)), x], x] + Dist[((b*e - a*f)*(d*e - 2*c*f))/(2*f^2), Int[1/(Sqrt[a + b*x^2]*Sqrt[
c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[(b*d*e - b*c*f - a*d*f)/(2*f^2), Int[Sqrt[e + f*x^2]/(Sqrt[a + b*x^
2]*Sqrt[c + d*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[(d*e - c*f)/c]

Rule 554

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[(Sqrt
[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(a*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]),
Subst[Int[Sqrt[1 - ((b*c - a*d)*x^2)/c]/Sqrt[1 - ((b*e - a*f)*x^2)/e], x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[
{a, b, c, d, e, f}, x]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 552

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[(Sqrt
[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(c*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]),
Subst[Int[1/(Sqrt[1 - ((b*c - a*d)*x^2)/c]*Sqrt[1 - ((b*e - a*f)*x^2)/e]), x], x, x/Sqrt[a + b*x^2]], x] /; Fr
eeQ[{a, b, c, d, e, f}, x]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 553

Int[Sqrt[(a_) + (b_.)*(x_)^2]/(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[(a*Sqrt
[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(c*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]),
Subst[Int[1/((1 - b*x^2)*Sqrt[1 - ((b*c - a*d)*x^2)/c]*Sqrt[1 - ((b*e - a*f)*x^2)/e]), x], x, x/Sqrt[a + b*x^2
]], x] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx &=\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{(a (b c-a d)) \int \frac{\sqrt{e+f x^2}}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \, dx}{2 b}+\frac{((b c-a d) (2 b e-a f)) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 b^2}+\frac{(b d e+b c f-a d f) \int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 b^2}\\ &=\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\left ((b c-a d) \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt{e+f x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(b e-a f) x^2}{e}}}{\sqrt{1-\frac{(b c-a d) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac{\left ((b c-a d) (2 b e-a f) \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{(b c-a d) x^2}{c}} \sqrt{1-\frac{(b e-a f) x^2}{e}}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 b^2 c \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt{e+f x^2}}+\frac{\left (a (b d e+b c f-a d f) \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-b x^2\right ) \sqrt{1-\frac{(b c-a d) x^2}{c}} \sqrt{1-\frac{(b e-a f) x^2}{e}}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 b^2 c \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt{e+f x^2}}\\ &=\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac{(b c-a d) \sqrt{e} (2 b e-a f) \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{a+b x^2}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt{e+f x^2}}+\frac{a (b d e+b c f-a d f) \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt{c} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt{e+f x^2}}\\ \end{align*}

Mathematica [A]  time = 1.62308, size = 512, normalized size = 0.95 \[ \frac{\sqrt{a+b x^2} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (\sqrt{e} \sqrt{a+b x^2} \sqrt{b c-a d} (2 b c-a d) \sqrt{b e-a f} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{e} \sqrt{a+b x^2}}\right ),\frac{b c e-a d e}{b c e-a c f}\right )+b^2 c x \left (e+f x^2\right ) \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}-b c \sqrt{e} \sqrt{a+b x^2} \sqrt{b c-a d} \sqrt{b e-a f} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{b c e-a d e}{b c e-a c f}\right )-a \sqrt{c} \sqrt{a+b x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b (c f+d e)) \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{b c e-a c f}{b c e-a d e}\right )\right )}{2 a b^2 \sqrt{c+d x^2} \sqrt{e+f x^2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*(b^2*c*Sqrt[b*c - a*d]*x*Sqrt[(a*(c + d*x^2))/(c*(a + b
*x^2))]*(e + f*x^2) - b*c*Sqrt[b*c - a*d]*Sqrt[e]*Sqrt[b*e - a*f]*Sqrt[a + b*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a +
b*x^2))]*EllipticE[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], (b*c*e - a*d*e)/(b*c*e - a*c*f)] +
Sqrt[b*c - a*d]*(2*b*c - a*d)*Sqrt[e]*Sqrt[b*e - a*f]*Sqrt[a + b*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*El
lipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], (b*c*e - a*d*e)/(b*c*e - a*c*f)] - a*Sqrt[c]*(a
*d*f - b*(d*e + c*f))*Sqrt[a + b*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticPi[(b*c)/(b*c - a*d), ArcS
in[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (b*c*e - a*c*f)/(b*c*e - a*d*e)]))/(2*a*b^2*Sqrt[b*c - a*d]
*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{d{x}^{2}+c}\sqrt{f{x}^{2}+e}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2),x)

[Out]

int((d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{\sqrt{b x^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/sqrt(b*x^2 + a), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{\sqrt{a + b x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**(1/2)*(f*x**2+e)**(1/2)/(b*x**2+a)**(1/2),x)

[Out]

Integral(sqrt(c + d*x**2)*sqrt(e + f*x**2)/sqrt(a + b*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{\sqrt{b x^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/sqrt(b*x^2 + a), x)